Let us set some global options for all code chunks in this document.

# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = FALSE,    
  # Disable warnings printed by R code chunks
  warning = FALSE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(grateful)

library(plotly)

We want to solve the fractional diffusion equation \[\begin{equation} \label{eq:maineq} \partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma, \end{equation}\] where \(u\) satisfies the Kirchhoff vertex conditions \[\begin{equation} \label{eq:Kcond} \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\} \end{equation}\]

Let us start by building a graph.

graph <- metric_graph$new(perform_merges = TRUE, 
                          tolerance = list(edge_edge = 1e-3, 
                                           vertex_vertex = 1e-3, 
                                           edge_vertex = 1e-3))
graph$plot()

graph$build_mesh(h = 0.05)

Let \(\alpha\in(0,2]\) and \(U_h^\tau\) denote the sequence of approximations of the solution to the weak form of problem \(\eqref{eq:maineq}\) at each time step on a mesh indexed by \(h\). Let \(z=0\) and \(U^0_h = P_hu_0\). For \(k=0,\dots, N-1\), \(U_h^{k+1}\in V_h\) solves the following scheme \[\begin{align} \label{system:fully_discrete_scheme} \langle\delta U_h^{k+1},\phi\rangle + \mathfrak{a}(U_h^{k+1},\phi) = \langle f^{k+1},\phi\rangle ,\quad\forall\phi\in V_h, \end{align}\] where \(f^{k+1} = \displaystyle\dfrac{1}{\tau}\int_{t_k}^{t^{k+1}}f(t)dt\). The solution can be represented as \[\begin{align*} U_h^k(s) = \sum_{j=1}^{N_h}u_j^k\psi_j(s) \end{align*}\] Replacing this into \(\eqref{system:fully_discrete_scheme}\) yields the following linear system \[\begin{align*} \sum_{j=1}^{N_h}u_j^{k+1}[(\psi_j,\psi_i)_{L_2(\Gamma)}+ \tau\mathfrak{a}(\psi_j,\psi_i)] = \sum_{j=1}^{N_h}u_j^{k}(\psi_j,\psi_i)_{L_2(\Gamma)}+\tau( f^{k+1},\psi_i)_{L_2(\Gamma)} \end{align*}\] for \(i = 1,\dots, N_h\). In matrix notation, \[\begin{align} \label{diff_eq_discrete} (C+\tau A)U^{k+1} = CU^k+\tau F^{k+1}, \end{align}\] where \(C\) has entries \(C_{ij} = (\psi_j,\psi_i)_{L_2(\Gamma)}\), \(A\) has entries \(A_{ij} = \mathfrak{a}(\psi_j,\psi_i)\), \(U^k\) has entries \(u_j^k\), and \(F^k\) has entries \(( f^{k},\psi_i)_{L_2(\Gamma)}\). By multiplying both sides by \(A^{-1}\) and considering its operator-based rational approximation \(P_\ell^{-1}P_r\), we arrive at \((P_rC+\tau P_\ell)U^{k+1} = P_r(CU^k+\tau F^{k+1})\).

# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
# Initial condition
U_0 <- 10*exp(-((x-4)^2 + (y-4)^2))
# Define the time step
time_step <- 0.1
# Define the right-hand side function
fun <- function(t) {return(sin(t)*((x-4)^2 - (y-4)^2))}
# Define the time discretization
time_seq <- seq(0,pi, by = time_step)
# Compute the right-hand side function at each time step
fun_mat <- do.call(cbind, lapply(time_seq, fun))
# Define the parameters
kappa <- 1
L <- kappa^2*C + G
alpha <- 0.8
beta <- alpha/2
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
funF <- C %*% fun_mat 
# Precompute the LHS matrix
LHS <- Pr %*% C + time_step * Pl
# Initialize U matrix to store solution at each time step
U_mat <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_mat[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  RHS <- Pr %*% (C %*% U_mat[, k] + time_step * funF[, k + 1])
  U_mat[, k + 1] <- as.matrix(solve(LHS, RHS))
}
# Plot the initial condition
p_ini <- graph$plot_function(X = U_0, 
                             vertex_size = 1, 
                             type = "plotly", 
                             edge_color = "black", 
                             edge_width = 3, 
                             line_color = "blue", 
                             line_width = 3)
p_ini
# Plot the movie of f
p_f <- graph$plot_movie(fun_mat)
p_f$x$layout$scene$xaxis$range <- range(x)
p_f$x$layout$scene$yaxis$range <- range(y)
p_f$x$layout$scene$zaxis$range <- range(fun_mat)
p_f
# Plot the movie of the solution
p_sol <- graph$plot_movie(U_mat)
p_sol$x$layout$scene$xaxis$range <- range(x)
p_sol$x$layout$scene$yaxis$range <- range(y)
p_sol$x$layout$scene$zaxis$range <- range(U_mat)
p_sol

References

cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.4.3 (R Core Team 2025) and the following R packages: htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 25.4.16 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.12.0.9012 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.48 (Xie 2014, 2015, 2024), MetricGraph v. 1.4.1.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), plotly v. 4.10.4 (Sievert 2020), rmarkdown v. 2.28 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.5.1.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

Aden-Buie, Garrick, and Matthew T. Warkentin. 2024. xaringanExtra: Extras and Extensions for xaringan Slides. https://CRAN.R-project.org/package=xaringanExtra.
Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier Luraschi, Kevin Ushey, Aron Atkins, et al. 2024. rmarkdown: Dynamic Documents for r. https://github.com/rstudio/rmarkdown.
Bachl, Fabian E., Finn Lindgren, David L. Borchers, and Janine B. Illian. 2019. inlabru: An R Package for Bayesian Spatial Modelling from Ecological Survey Data.” Methods in Ecology and Evolution 10: 760–66. https://doi.org/10.1111/2041-210X.13168.
Bakka, Haakon, Håvard Rue, Geir-Arne Fuglstad, Andrea I. Riebler, David Bolin, Janine Illian, Elias Krainski, Daniel P. Simpson, and Finn K. Lindgren. 2018. “Spatial Modelling with INLA: A Review.” WIRES (Invited Extended Review) xx (Feb): xx–. http://arxiv.org/abs/1802.06350.
Bolin, David, and Kristin Kirchner. 2020. “The Rational SPDE Approach for Gaussian Random Fields with General Smoothness.” Journal of Computational and Graphical Statistics 29 (2): 274–85. https://doi.org/10.1080/10618600.2019.1665537.
Bolin, David, Mihály Kovács, Vivek Kumar, and Alexandre B. Simas. 2024. “Regularity and Numerical Approximation of Fractional Elliptic Differential Equations on Compact Metric Graphs.” Mathematics of Computation 93 (349): 2439–72. https://doi.org/10.1090/mcom/3929.
Bolin, David, and Alexandre B. Simas. 2023. rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations. https://CRAN.R-project.org/package=rSPDE.
Bolin, David, Alexandre B. Simas, and Jonas Wallin. 2023a. MetricGraph: Random Fields on Metric Graphs. https://CRAN.R-project.org/package=MetricGraph.
———. 2023b. “Statistical Inference for Gaussian Whittle-Matérn Fields on Metric Graphs.” arXiv Preprint arXiv:2304.10372. https://doi.org/10.48550/arXiv.2304.10372.
———. 2024. “Gaussian Whittle-Matérn Fields on Metric Graphs.” Bernoulli 30 (2): 1611–39. https://doi.org/10.3150/23-BEJ1647.
———. 2025. “Markov Properties of Gaussian Random Fields on Compact Metric Graphs.” Bernoulli. https://doi.org/10.48550/arXiv.2304.03190.
Bolin, David, Alexandre B. Simas, and Zhen Xiong. 2024. “Covariance-Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference.” Journal of Computational and Graphical Statistics 33 (1): 64–74. https://doi.org/10.1080/10618600.2023.2231051.
Cheng, Joe, Carson Sievert, Barret Schloerke, Winston Chang, Yihui Xie, and Jeff Allen. 2024. htmltools: Tools for HTML. https://CRAN.R-project.org/package=htmltools.
De Coninck, Arne, Bernard De Baets, Drosos Kourounis, Fabio Verbosio, Olaf Schenk, Steven Maenhout, and Jan Fostier. 2016. Needles: Toward Large-Scale Genomic Prediction with Marker-by-Environment Interaction.” Genetics 203 (1): 543–55. https://doi.org/10.1534/genetics.115.179887.
Kourounis, D., A. Fuchs, and O. Schenk. 2018. “Towards the Next Generation of Multiperiod Optimal Power Flow Solvers.” IEEE Transactions on Power Systems PP (99): 1–10. https://doi.org/10.1109/TPWRS.2017.2789187.
Lindgren, Finn, and Håvard Rue. 2015. “Bayesian Spatial Modelling with R-INLA.” Journal of Statistical Software 63 (19): 1–25. http://www.jstatsoft.org/v63/i19/.
Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach (with Discussion).” Journal of the Royal Statistical Society B 73 (4): 423–98.
Martins, Thiago G., Daniel Simpson, Finn Lindgren, and Håvard Rue. 2013. “Bayesian Computing with INLA: New Features.” Computational Statistics and Data Analysis 67: 68–83.
R Core Team. 2025. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.
Rue, Håvard, Sara Martino, and Nicholas Chopin. 2009. “Approximate Bayesian Inference for Latent Gaussian Models Using Integrated Nested Laplace Approximations (with Discussion).” Journal of the Royal Statistical Society B 71: 319–92.
Rue, Håvard, Andrea I. Riebler, Sigrunn H. Sørbye, Janine B. Illian, Daniel P. Simpson, and Finn K. Lindgren. 2017. “Bayesian Computing with INLA: A Review.” Annual Reviews of Statistics and Its Applications 4 (March): 395–421. http://arxiv.org/abs/1604.00860.
Sievert, Carson. 2020. Interactive Web-Based Data Visualization with r, Plotly, and Shiny. Chapman; Hall/CRC. https://plotly-r.com.
Verbosio, Fabio, Arne De Coninck, Drosos Kourounis, and Olaf Schenk. 2017. “Enhancing the Scalability of Selected Inversion Factorization Algorithms in Genomic Prediction.” Journal of Computational Science 22 (Supplement C): 99–108. https://doi.org/10.1016/j.jocs.2017.08.013.
Xie, Yihui. 2014. knitr: A Comprehensive Tool for Reproducible Research in R.” In Implementing Reproducible Computational Research, edited by Victoria Stodden, Friedrich Leisch, and Roger D. Peng. Chapman; Hall/CRC.
———. 2015. Dynamic Documents with R and Knitr. 2nd ed. Boca Raton, Florida: Chapman; Hall/CRC. https://yihui.org/knitr/.
———. 2024. knitr: A General-Purpose Package for Dynamic Report Generation in r. https://yihui.org/knitr/.
Xie, Yihui, J. J. Allaire, and Garrett Grolemund. 2018. R Markdown: The Definitive Guide. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown.
Xie, Yihui, Christophe Dervieux, and Emily Riederer. 2020. R Markdown Cookbook. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown-cookbook.
Yuan, Yuan, Bachl, Fabian E., Lindgren, Finn, Borchers, et al. 2017. “Point Process Models for Spatio-Temporal Distance Sampling Data from a Large-Scale Survey of Blue Whales.” Ann. Appl. Stat. 11 (4): 2270–97. https://doi.org/10.1214/17-AOAS1078.
---
title: "Solving a parabolic equation"
date: "Created: 20-04-2025. Last modified: `r format(Sys.time(), '%d-%m-%Y.')`"
output:
  html_document:
    mathjax: "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
    highlight: pygments
    theme: flatly
    code_folding: show # class.source = "fold-hide" to hide code and add a button to show it
    df_print: paged
    toc: true
    toc_float:
      collapsed: true
      smooth_scroll: true
    number_sections: false
    fig_caption: true
    code_download: true
always_allow_html: true
bibliography: 
  - references.bib
  - grateful-refs.bib
header-includes:
  - \newcommand{\ar}{\mathbb{R}}
  - \newcommand{\llav}[1]{\left\{#1\right\}}
  - \newcommand{\pare}[1]{\left(#1\right)}
  - \newcommand{\Ncal}{\mathcal{N}}
  - \newcommand{\Vcal}{\mathcal{V}}
  - \newcommand{\Ecal}{\mathcal{E}}
  - \newcommand{\Wcal}{\mathcal{W}}
---

```{r xaringanExtra-clipboard, echo = FALSE}
htmltools::tagList(
  xaringanExtra::use_clipboard(
    button_text = "<i class=\"fa-solid fa-clipboard\" style=\"color: #00008B\"></i>",
    success_text = "<i class=\"fa fa-check\" style=\"color: #90BE6D\"></i>",
    error_text = "<i class=\"fa fa-times-circle\" style=\"color: #F94144\"></i>"
  ),
  rmarkdown::html_dependency_font_awesome()
)
```


```{css, echo = FALSE}
body .main-container {
  max-width: 100% !important;
  width: 100% !important;
}
body {
  max-width: 100% !important;
}

body, td {
   font-size: 16px;
}
code.r{
  font-size: 14px;
}
pre {
  font-size: 14px
}
.custom-box {
  background-color: #f5f7fa; /* Light grey-blue background */
  border-color: #e1e8ed; /* Light border color */
  color: #2c3e50; /* Dark text color */
  padding: 15px; /* Padding inside the box */
  border-radius: 5px; /* Rounded corners */
  margin-bottom: 20px; /* Spacing below the box */
}
.caption {
  margin: auto;
  text-align: center;
  margin-bottom: 20px; /* Spacing below the box */
}
```


Let us set some global options for all code chunks in this document.


```{r}
# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = FALSE,    
  # Disable warnings printed by R code chunks
  warning = FALSE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
```




```{r}
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)
library(inlabru)
library(rSPDE)
library(MetricGraph)
library(grateful)

library(plotly)
```


We want to solve the fractional diffusion equation
\begin{equation}
\label{eq:maineq}
    \partial_t u+(\kappa^2-\Delta_\Gamma)^{\frac{\alpha}{2}} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma,
\end{equation}
where $u$ satisfies the Kirchhoff vertex conditions
\begin{equation}
\label{eq:Kcond}
    \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\}
\end{equation}

Let us start by building a graph.

```{r}
graph <- metric_graph$new(perform_merges = TRUE, 
                          tolerance = list(edge_edge = 1e-3, 
                                           vertex_vertex = 1e-3, 
                                           edge_vertex = 1e-3))
graph$plot()
graph$build_mesh(h = 0.05)
```

Let $\alpha\in(0,2]$ and $U_h^\tau$ denote the sequence of approximations of the solution to the weak form of problem \eqref{eq:maineq} at each time step on a mesh indexed by $h$. Let $z=0$ and $U^0_h = P_hu_0$. For $k=0,\dots, N-1$, $U_h^{k+1}\in V_h$ solves the following scheme
\begin{align}
\label{system:fully_discrete_scheme}
        \langle\delta U_h^{k+1},\phi\rangle + \mathfrak{a}(U_h^{k+1},\phi) = \langle f^{k+1},\phi\rangle ,\quad\forall\phi\in V_h,
\end{align}
where $f^{k+1} = \displaystyle\dfrac{1}{\tau}\int_{t_k}^{t^{k+1}}f(t)dt$.
The solution can be represented as 
\begin{align*}
    U_h^k(s) =  \sum_{j=1}^{N_h}u_j^k\psi_j(s)
\end{align*}
Replacing this into \eqref{system:fully_discrete_scheme} yields the following linear system
\begin{align*}
    \sum_{j=1}^{N_h}u_j^{k+1}[(\psi_j,\psi_i)_{L_2(\Gamma)}+ \tau\mathfrak{a}(\psi_j,\psi_i)] = \sum_{j=1}^{N_h}u_j^{k}(\psi_j,\psi_i)_{L_2(\Gamma)}+\tau( f^{k+1},\psi_i)_{L_2(\Gamma)}
\end{align*}
for $i = 1,\dots, N_h$. In matrix notation,
\begin{align}
\label{diff_eq_discrete}
    (C+\tau A)U^{k+1} = CU^k+\tau F^{k+1},
\end{align}
where $C$ has entries $C_{ij} = (\psi_j,\psi_i)_{L_2(\Gamma)}$, $A$ has entries $A_{ij} = \mathfrak{a}(\psi_j,\psi_i)$, $U^k$ has entries $u_j^k$, and $F^k$ has entries $( f^{k},\psi_i)_{L_2(\Gamma)}$. By multiplying both sides by $A^{-1}$ and considering its operator-based rational approximation $P_\ell^{-1}P_r$, we arrive at $(P_rC+\tau P_\ell)U^{k+1} = P_r(CU^k+\tau F^{k+1})$.

```{r}
# Compute the FEM matrices
graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
# Initial condition
U_0 <- 10*exp(-((x-4)^2 + (y-4)^2))
# Define the time step
time_step <- 0.1
# Define the right-hand side function
fun <- function(t) {return(sin(t)*((x-4)^2 - (y-4)^2))}
# Define the time discretization
time_seq <- seq(0,pi, by = time_step)
# Compute the right-hand side function at each time step
fun_mat <- do.call(cbind, lapply(time_seq, fun))
# Define the parameters
kappa <- 1
L <- kappa^2*C + G
alpha <- 0.8
beta <- alpha/2
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = 1)
Pl <- op$Pl
Pr <- op$Pr
funF <- C %*% fun_mat 
```


```{r}
# Precompute the LHS matrix
LHS <- Pr %*% C + time_step * Pl
# Initialize U matrix to store solution at each time step
U_mat <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_mat[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  RHS <- Pr %*% (C %*% U_mat[, k] + time_step * funF[, k + 1])
  U_mat[, k + 1] <- as.matrix(solve(LHS, RHS))
}
```


```{r}
# Plot the initial condition
p_ini <- graph$plot_function(X = U_0, 
                             vertex_size = 1, 
                             type = "plotly", 
                             edge_color = "black", 
                             edge_width = 3, 
                             line_color = "blue", 
                             line_width = 3)
p_ini
# Plot the movie of f
p_f <- graph$plot_movie(fun_mat)
p_f$x$layout$scene$xaxis$range <- range(x)
p_f$x$layout$scene$yaxis$range <- range(y)
p_f$x$layout$scene$zaxis$range <- range(fun_mat)
p_f
# Plot the movie of the solution
p_sol <- graph$plot_movie(U_mat)
p_sol$x$layout$scene$xaxis$range <- range(x)
p_sol$x$layout$scene$yaxis$range <- range(y)
p_sol$x$layout$scene$zaxis$range <- range(U_mat)
p_sol
```



# References

```{r}
cite_packages(output = "paragraph", out.dir = ".")
```
